466 lines
17 KiB
C++
466 lines
17 KiB
C++
// MFEM Example 31
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//
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// Compile with: make ex31
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//
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// Sample runs: ex31 -m ../data/inline-segment.mesh -o 2
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// ex31 -m ../data/hexagon.mesh -o 2
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// ex31 -m ../data/star.mesh -o 2
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// ex31 -m ../data/fichera.mesh -o 3 -r 1
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// ex31 -m ../data/square-disc-nurbs.mesh -o 3
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// ex31 -m ../data/amr-quad.mesh -o 2 -r 1
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// ex31 -m ../data/amr-hex.mesh -r 1
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//
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// Description: This example code solves a simple electromagnetic diffusion
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// problem corresponding to the second order definite Maxwell
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// equation curl curl E + sigma E = f with boundary condition
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// E x n = <given tangential field>. In this example sigma is an
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// anisotropic 3x3 tensor. Here, we use a given exact solution E
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// and compute the corresponding r.h.s. f. We discretize with
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// Nedelec finite elements in 1D, 2D, or 3D.
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//
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// The example demonstrates the use of restricted H(curl) finite
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// element spaces with the curl-curl and the (vector finite
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// element) mass bilinear form, as well as the computation of
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// discretization error when the exact solution is known. These
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// restricted spaces allow the solution of 1D or 2D
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// electromagnetic problems which involve 3D field vectors. Such
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// problems arise in plasma physics and crystallography.
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//
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// We recommend viewing example 3 before viewing this example.
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#include "mfem.hpp"
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#include <fstream>
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#include <iostream>
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using namespace std;
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using namespace mfem;
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// Exact solution, E, and r.h.s., f. See below for implementation.
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void E_exact(const Vector &, Vector &);
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void CurlE_exact(const Vector &, Vector &);
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void f_exact(const Vector &, Vector &);
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real_t freq = 1.0, kappa;
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int dim;
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int main(int argc, char *argv[])
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{
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// 1. Parse command-line options.
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const char *mesh_file = "../data/inline-quad.mesh";
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int ref_levels = 2;
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int order = 1;
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bool visualization = 1;
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OptionsParser args(argc, argv);
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args.AddOption(&mesh_file, "-m", "--mesh",
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"Mesh file to use.");
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args.AddOption(&ref_levels, "-r", "--refine",
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"Number of times to refine the mesh uniformly.");
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args.AddOption(&order, "-o", "--order",
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"Finite element order (polynomial degree).");
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args.AddOption(&freq, "-f", "--frequency", "Set the frequency for the exact"
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" solution.");
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args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
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"--no-visualization",
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"Enable or disable GLVis visualization.");
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args.ParseCheck();
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kappa = freq * M_PI;
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// 2. Read the mesh from the given mesh file. We can handle triangular,
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// quadrilateral, or mixed meshes with the same code.
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Mesh mesh(mesh_file, 1, 1);
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dim = mesh.Dimension();
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// 3. Refine the mesh to increase the resolution. In this example we do
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// 'ref_levels' of uniform refinement (2 by default, or specified on
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// the command line with -r).
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for (int lev = 0; lev < ref_levels; lev++)
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{
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mesh.UniformRefinement();
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}
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// 4. Define a finite element space on the mesh. Here we use the Nedelec
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// finite elements of the specified order restricted to 1D, 2D, or 3D
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// depending on the dimension of the given mesh file.
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FiniteElementCollection *fec = NULL;
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if (dim == 1)
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{
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fec = new ND_R1D_FECollection(order, dim);
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}
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else if (dim == 2)
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{
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fec = new ND_R2D_FECollection(order, dim);
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}
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else
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{
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fec = new ND_FECollection(order, dim);
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}
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FiniteElementSpace fespace(&mesh, fec);
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int size = fespace.GetTrueVSize();
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cout << "Number of H(Curl) unknowns: " << size << endl;
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// 5. Determine the list of true essential boundary dofs. In this example,
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// the boundary conditions are defined by marking all the boundary
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// attributes from the mesh as essential (Dirichlet) and converting them
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// to a list of true dofs.
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Array<int> ess_tdof_list;
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if (mesh.bdr_attributes.Size())
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{
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Array<int> ess_bdr(mesh.bdr_attributes.Max());
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ess_bdr = 1;
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fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
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}
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// 6. Set up the linear form b(.) which corresponds to the right-hand side
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// of the FEM linear system, which in this case is (f,phi_i) where f is
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// given by the function f_exact and phi_i are the basis functions in
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// the finite element fespace.
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VectorFunctionCoefficient f(3, f_exact);
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LinearForm b(&fespace);
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b.AddDomainIntegrator(new VectorFEDomainLFIntegrator(f));
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b.Assemble();
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// 7. Define the solution vector x as a finite element grid function
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// corresponding to fespace. Initialize x by projecting the exact
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// solution. Note that only values from the boundary edges will be used
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// when eliminating the non-homogeneous boundary condition to modify the
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// r.h.s. vector b.
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GridFunction sol(&fespace);
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VectorFunctionCoefficient E(3, E_exact);
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VectorFunctionCoefficient CurlE(3, CurlE_exact);
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sol.ProjectCoefficient(E);
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// 8. Set up the bilinear form corresponding to the EM diffusion operator
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// curl muinv curl + sigma I, by adding the curl-curl and the mass domain
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// integrators.
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DenseMatrix sigmaMat(3);
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sigmaMat(0,0) = 2.0; sigmaMat(1,1) = 2.0; sigmaMat(2,2) = 2.0;
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sigmaMat(0,2) = 0.0; sigmaMat(2,0) = 0.0;
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sigmaMat(0,1) = M_SQRT1_2; sigmaMat(1,0) = M_SQRT1_2; // 1/sqrt(2) in cmath
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sigmaMat(1,2) = M_SQRT1_2; sigmaMat(2,1) = M_SQRT1_2;
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ConstantCoefficient muinv(1.0);
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MatrixConstantCoefficient sigma(sigmaMat);
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BilinearForm a(&fespace);
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a.AddDomainIntegrator(new CurlCurlIntegrator(muinv));
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a.AddDomainIntegrator(new VectorFEMassIntegrator(sigma));
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// 9. Assemble the bilinear form and the corresponding linear system,
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// applying any necessary transformations such as: eliminating boundary
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// conditions, applying conforming constraints for non-conforming AMR,
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// etc.
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a.Assemble();
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OperatorPtr A;
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Vector B, X;
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a.FormLinearSystem(ess_tdof_list, sol, b, A, X, B);
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// 10. Solve the system A X = B.
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#ifndef MFEM_USE_SUITESPARSE
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// 11. Define a simple symmetric Gauss-Seidel preconditioner and use it to
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// solve the system Ax=b with PCG.
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GSSmoother M((SparseMatrix&)(*A));
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PCG(*A, M, B, X, 1, 500, 1e-12, 0.0);
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#else
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// 11. If MFEM was compiled with SuiteSparse, use UMFPACK to solve the
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// system.
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UMFPackSolver umf_solver;
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umf_solver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS;
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umf_solver.SetOperator(*A);
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umf_solver.Mult(B, X);
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#endif
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// 12. Recover the solution as a finite element grid function.
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a.RecoverFEMSolution(X, b, sol);
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// 13. Compute and print the H(Curl) norm of the error.
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{
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real_t error = sol.ComputeHCurlError(&E, &CurlE);
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cout << "\n|| E_h - E ||_{H(Curl)} = " << error << '\n' << endl;
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}
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// 14. Save the refined mesh and the solution. This output can be viewed
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// later using GLVis: "glvis -m refined.mesh -g sol.gf".
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{
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ofstream mesh_ofs("refined.mesh");
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mesh_ofs.precision(8);
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mesh.Print(mesh_ofs);
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ofstream sol_ofs("sol.gf");
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sol_ofs.precision(8);
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sol.Save(sol_ofs);
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}
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// 15. Send the solution by socket to a GLVis server.
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if (visualization)
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{
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char vishost[] = "localhost";
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int visport = 19916;
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VectorGridFunctionCoefficient solCoef(&sol);
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CurlGridFunctionCoefficient dsolCoef(&sol);
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if (dim ==1)
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{
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socketstream x_sock(vishost, visport);
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socketstream y_sock(vishost, visport);
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socketstream z_sock(vishost, visport);
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socketstream dy_sock(vishost, visport);
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socketstream dz_sock(vishost, visport);
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x_sock.precision(8);
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y_sock.precision(8);
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z_sock.precision(8);
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dy_sock.precision(8);
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dz_sock.precision(8);
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Vector xVec(3); xVec = 0.0; xVec(0) = 1;
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Vector yVec(3); yVec = 0.0; yVec(1) = 1;
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Vector zVec(3); zVec = 0.0; zVec(2) = 1;
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VectorConstantCoefficient xVecCoef(xVec);
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VectorConstantCoefficient yVecCoef(yVec);
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VectorConstantCoefficient zVecCoef(zVec);
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H1_FECollection fec_h1(order, dim);
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L2_FECollection fec_l2(order-1, dim);
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FiniteElementSpace fes_h1(&mesh, &fec_h1);
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FiniteElementSpace fes_l2(&mesh, &fec_l2);
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GridFunction xComp(&fes_l2);
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GridFunction yComp(&fes_h1);
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GridFunction zComp(&fes_h1);
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GridFunction dyComp(&fes_l2);
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GridFunction dzComp(&fes_l2);
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InnerProductCoefficient xCoef(xVecCoef, solCoef);
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InnerProductCoefficient yCoef(yVecCoef, solCoef);
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InnerProductCoefficient zCoef(zVecCoef, solCoef);
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xComp.ProjectCoefficient(xCoef);
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yComp.ProjectCoefficient(yCoef);
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zComp.ProjectCoefficient(zCoef);
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x_sock << "solution\n" << mesh << xComp << flush
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<< "window_title 'X component'" << endl;
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y_sock << "solution\n" << mesh << yComp << flush
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<< "window_geometry 403 0 400 350 "
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<< "window_title 'Y component'" << endl;
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z_sock << "solution\n" << mesh << zComp << flush
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<< "window_geometry 806 0 400 350 "
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<< "window_title 'Z component'" << endl;
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InnerProductCoefficient dyCoef(yVecCoef, dsolCoef);
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InnerProductCoefficient dzCoef(zVecCoef, dsolCoef);
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dyComp.ProjectCoefficient(dyCoef);
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dzComp.ProjectCoefficient(dzCoef);
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dy_sock << "solution\n" << mesh << dyComp << flush
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<< "window_geometry 403 375 400 350 "
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<< "window_title 'Y component of Curl'" << endl;
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dz_sock << "solution\n" << mesh << dzComp << flush
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<< "window_geometry 806 375 400 350 "
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<< "window_title 'Z component of Curl'" << endl;
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}
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else if (dim == 2)
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{
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socketstream xy_sock(vishost, visport);
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socketstream z_sock(vishost, visport);
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socketstream dxy_sock(vishost, visport);
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socketstream dz_sock(vishost, visport);
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DenseMatrix xyMat(2,3); xyMat = 0.0;
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xyMat(0,0) = 1.0; xyMat(1,1) = 1.0;
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MatrixConstantCoefficient xyMatCoef(xyMat);
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Vector zVec(3); zVec = 0.0; zVec(2) = 1;
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VectorConstantCoefficient zVecCoef(zVec);
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MatrixVectorProductCoefficient xyCoef(xyMatCoef, solCoef);
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InnerProductCoefficient zCoef(zVecCoef, solCoef);
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H1_FECollection fec_h1(order, dim);
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ND_FECollection fec_nd(order, dim);
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RT_FECollection fec_rt(order-1, dim);
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L2_FECollection fec_l2(order-1, dim);
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FiniteElementSpace fes_h1(&mesh, &fec_h1);
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FiniteElementSpace fes_nd(&mesh, &fec_nd);
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FiniteElementSpace fes_rt(&mesh, &fec_rt);
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FiniteElementSpace fes_l2(&mesh, &fec_l2);
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GridFunction xyComp(&fes_nd);
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GridFunction zComp(&fes_h1);
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GridFunction dxyComp(&fes_rt);
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GridFunction dzComp(&fes_l2);
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xyComp.ProjectCoefficient(xyCoef);
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zComp.ProjectCoefficient(zCoef);
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xy_sock.precision(8);
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xy_sock << "solution\n" << mesh << xyComp
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<< "window_title 'XY components'\n" << flush;
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z_sock << "solution\n" << mesh << zComp << flush
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<< "window_geometry 403 0 400 350 "
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<< "window_title 'Z component'" << endl;
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MatrixVectorProductCoefficient dxyCoef(xyMatCoef, dsolCoef);
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InnerProductCoefficient dzCoef(zVecCoef, dsolCoef);
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dxyComp.ProjectCoefficient(dxyCoef);
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dzComp.ProjectCoefficient(dzCoef);
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dxy_sock << "solution\n" << mesh << dxyComp << flush
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<< "window_geometry 0 375 400 350 "
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<< "window_title 'XY components of Curl'" << endl;
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dz_sock << "solution\n" << mesh << dzComp << flush
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<< "window_geometry 403 375 400 350 "
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<< "window_title 'Z component of Curl'" << endl;
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}
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else
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{
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socketstream sol_sock(vishost, visport);
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socketstream dsol_sock(vishost, visport);
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RT_FECollection fec_rt(order-1, dim);
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FiniteElementSpace fes_rt(&mesh, &fec_rt);
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GridFunction dsol(&fes_rt);
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dsol.ProjectCoefficient(dsolCoef);
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sol_sock.precision(8);
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sol_sock << "solution\n" << mesh << sol
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<< "window_title 'Solution'" << flush << endl;
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dsol_sock << "solution\n" << mesh << dsol << flush
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<< "window_geometry 0 375 400 350 "
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<< "window_title 'Curl of solution'" << endl;
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}
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}
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// 16. Free the used memory.
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delete fec;
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return 0;
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}
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void E_exact(const Vector &x, Vector &E)
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{
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if (dim == 1)
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{
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E(0) = 1.1 * sin(kappa * x(0) + 0.0 * M_PI);
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E(1) = 1.2 * sin(kappa * x(0) + 0.4 * M_PI);
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E(2) = 1.3 * sin(kappa * x(0) + 0.9 * M_PI);
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}
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else if (dim == 2)
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{
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E(0) = 1.1 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
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E(1) = 1.2 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
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E(2) = 1.3 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
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}
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else
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{
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E(0) = 1.1 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
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E(1) = 1.2 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
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E(2) = 1.3 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
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E *= cos(kappa * x(2));
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}
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}
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void CurlE_exact(const Vector &x, Vector &dE)
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{
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if (dim == 1)
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{
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real_t c4 = cos(kappa * x(0) + 0.4 * M_PI);
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real_t c9 = cos(kappa * x(0) + 0.9 * M_PI);
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dE(0) = 0.0;
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dE(1) = -1.3 * c9;
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dE(2) = 1.2 * c4;
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dE *= kappa;
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}
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else if (dim == 2)
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{
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real_t c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
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real_t c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
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real_t c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
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dE(0) = 1.3 * c9;
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dE(1) = -1.3 * c9;
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dE(2) = 1.2 * c4 - 1.1 * c0;
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dE *= kappa * M_SQRT1_2;
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}
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else
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{
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real_t s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
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real_t c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
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real_t s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
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real_t c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
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real_t c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
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real_t sk = sin(kappa * x(2));
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real_t ck = cos(kappa * x(2));
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dE(0) = 1.2 * s4 * sk + 1.3 * M_SQRT1_2 * c9 * ck;
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dE(1) = -1.1 * s0 * sk - 1.3 * M_SQRT1_2 * c9 * ck;
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dE(2) = -M_SQRT1_2 * (1.1 * c0 - 1.2 * c4) * ck;
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dE *= kappa;
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}
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}
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void f_exact(const Vector &x, Vector &f)
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{
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if (dim == 1)
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{
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real_t s0 = sin(kappa * x(0) + 0.0 * M_PI);
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real_t s4 = sin(kappa * x(0) + 0.4 * M_PI);
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real_t s9 = sin(kappa * x(0) + 0.9 * M_PI);
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f(0) = 2.2 * s0 + 1.2 * M_SQRT1_2 * s4;
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f(1) = 1.2 * (2.0 + kappa * kappa) * s4 +
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M_SQRT1_2 * (1.1 * s0 + 1.3 * s9);
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f(2) = 1.3 * (2.0 + kappa * kappa) * s9 + 1.2 * M_SQRT1_2 * s4;
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}
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else if (dim == 2)
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{
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real_t s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
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real_t s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
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real_t s9 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
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f(0) = 0.55 * (4.0 + kappa * kappa) * s0 +
|
|
0.6 * (M_SQRT2 - kappa * kappa) * s4;
|
|
f(1) = 0.55 * (M_SQRT2 - kappa * kappa) * s0 +
|
|
0.6 * (4.0 + kappa * kappa) * s4 +
|
|
0.65 * M_SQRT2 * s9;
|
|
f(2) = 0.6 * M_SQRT2 * s4 + 1.3 * (2.0 + kappa * kappa) * s9;
|
|
}
|
|
else
|
|
{
|
|
real_t s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
|
|
real_t c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
|
|
real_t s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
|
|
real_t c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
|
|
real_t s9 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
|
|
real_t c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
|
|
real_t sk = sin(kappa * x(2));
|
|
real_t ck = cos(kappa * x(2));
|
|
|
|
f(0) = 0.55 * (4.0 + 3.0 * kappa * kappa) * s0 * ck +
|
|
0.6 * (M_SQRT2 - kappa * kappa) * s4 * ck -
|
|
0.65 * M_SQRT2 * kappa * kappa * c9 * sk;
|
|
|
|
f(1) = 0.55 * (M_SQRT2 - kappa * kappa) * s0 * ck +
|
|
0.6 * (4.0 + 3.0 * kappa * kappa) * s4 * ck +
|
|
0.65 * M_SQRT2 * s9 * ck -
|
|
0.65 * M_SQRT2 * kappa * kappa * c9 * sk;
|
|
|
|
f(2) = 0.6 * M_SQRT2 * s4 * ck -
|
|
M_SQRT2 * kappa * kappa * (0.55 * c0 + 0.6 * c4) * sk
|
|
+ 1.3 * (2.0 + kappa * kappa) * s9 * ck;
|
|
}
|
|
}
|